By introducing trigonometric-type bilinear operators, we propose two novel discrete integrable equations that can be viewed as variants of the Leznov lattice and the differential-difference Kadomtsev-Petviashvili (D Delta KP) equations. It turns out that both equations admit various solutions, including general Grammian determinant, soliton, lump, rogue wave, and breather solutions, which are expressed by explicit and closed forms. Moreover, g-periodic wave solutions are also constructed in terms of Riemann theta function. Numerical three-periodic wave solutions are successfully computed by using a deep neural network. Finally, we construct a continuum limit, through which we reveal clear links between the variant D Delta KP equation and the Kadomtsev-Petviashvili-I (KPI) equation from both the equation and solution perspectives.

Publication:

PHYSICA D-NONLINEAR PHENOMENA

http://dx.doi.org/10.1016/j.physd.2025.134831

Author:

Ya-Jie Liu

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Corresponding author

E-mail address: liuyajie@lsec.cc.ac.cn

Hui Alan Wang

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

Xing-Biao Hu

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

Ying-Nan Zhang

Ministry of Education Key Laboratory of NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, Jiangsu 210023, China